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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Differentiation > Symbolic differentiation > With respect to nu





http://functions.wolfram.com/07.07.20.0008.02









  


  








Input Form





D[LegendreP[\[Nu], z], {\[Nu], m}] == Sum[(1/k!^2) ((1 - z)/2)^k Sum[Binomial[m, j] Sum[StirlingS1[k, i] Pochhammer[i - j + 1, j] \[Nu]^(i - j) Sum[(-1)^r StirlingS1[k, r] Pochhammer[r - m + j + 1, m - j] (\[Nu] + 1)^(r - m + j), {r, 1, k}], {i, 1, k}], {j, 0, m}], {k, 0, Infinity}] /; Abs[(1 - z)/2] < 1 && Element[m, Integers] && m >= 0








Standard Form





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MathML Form







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</mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;m&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> k </mi> <mrow> <mo> ( </mo> <mi> i </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;i&quot;, &quot;-&quot;, &quot;j&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; 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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "m_"]], "}"]]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["i", "-", "j", "+", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox["\[Nu]", RowBox[List["i", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["r", "-", "m", "+", "j", "+", "1"]], ",", RowBox[List["m", "-", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["r", "-", "m", "+", "j"]]]]]]]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29